Convolutions and integral equations weighted by multi-dimensional Hermite functions

Abstract

In this paper, we study the solvability of a very general class of integral equations whose kernel depends on four different functions. Necessary and sufficient conditions for the unique solvability of such integral equations are obtained. To achieve such a goal, the main technique consists in introducing eight new convolutions weighted by multi-dimensional Hermite functions and using them as convolutions somehow associated with our integral equations. In this way, two Young-type inequalities are also obtained.